HP -15C Advanced Functions Handbook page 60

58 Section 2: Working With [F\n tails, like that of
f(x), can be truncated
degrading the accuracy or speed of integration. But g(x) has too
wide a tail to ignore when calculating
t
s(x)dx
L
if £is large.
For such functions, a substitution like x = a + b tan u works well,
where a lies within the graph's main "body" and b is roughly its
width. Doing this for f(x) from above with a = 0 and b = I gives
/
t rtan~
l
t
f(x) dx = I e'
ian
\l + tan
2
u)du,
J
J
0
which is calculated readily even with t as large as 10
10
. Using the
same substitution with g( x), values near a = 0 and b = 10~
5
provide
good results.
This example involves subdividing the interval of integration.
Although a function may have features that look extreme over the
entire interval of integration, over portions of that interval the
function may look more well-behaved. Subdividing the interval of
integration works best when combined with appropriate substitu-
tions. Consider the integral
I dx/(l+x
&4
) = dx/(l+x
M
) +
J
0
J
0
J
1
r l r l
=) dx/(l + x
M
)+J u
6
'
2
du/(u
C
l
= /
J
0
rl
1 (x
6
J
o
(
These steps use the substitutions x = I/ u and x = v
l/8
and some
algebraic manipulation. Although the original integral is
improper, the last integral is easily handled by [7T|. In fact, by
separating the constant term from the integral, you obtain (using
| SCI 1 8) an answer with 13 significant digits: